1
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Benisty H. Growth Quakes and Stasis Using Iterations of Inflating Complex Random Matrices. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1507. [PMID: 37998199 PMCID: PMC10670579 DOI: 10.3390/e25111507] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2023] [Revised: 10/25/2023] [Accepted: 10/26/2023] [Indexed: 11/25/2023]
Abstract
I extend to the case of complex matrices, rather than the case of real matrices as in a prior study, a method of iterating the operation of an "inflating random matrix" onto a state vector to describe complex growing systems. I show that the process also describes in this complex case a punctuated growth with quakes and stasis. I assess that under one such inflation step, the vector will shift to a really different one (quakes) only if the inflated matrix has sufficiently dominant new eigenvectors. The vector shall prefer stasis (a similar vector) otherwise, similar to the real-valued matrices discussed in a prior study. Specifically, in order to extend the model relevance, I assess that under various update schemes of the system's representative vector, the bimodal distribution of the changes of the dominant eigenvalue remains the core concept. Overall, I contend that the punctuations may appropriately address the issue of growth in systems combining a large weight of history and some sudden quake occurrences, such as economic systems or ecological systems, with the advantage that unpaired complex eigenvalues provide more degrees of freedom to suit real systems. Furthermore, random matrices could be the right meeting point for exerting thermodynamic analogies in a reasonably agnostic manner in such rich contexts, taking into account the profusion of items (individuals, species, goods, etc.) and their networked, tangled interactions 50+ years after their seminal use in R.M. May's famous "interaction induced instability" paradigm. Finally, I suggest that non-ergodic tools could be further applied for tracking the specifics of large-scale evolution paths and for checking the model's relevance to the domains mentioned above.
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Affiliation(s)
- Henri Benisty
- Laboratoire Charles Fabry, IOGS, Université Paris-Saclay, 2 Av. Fresnel, 91120 Palaiseau, France; ; Tel.: +33-1-6453-3286
- LIED Laboratory, Université Paris Cité, 5 rue Marie-Andrée Lagroua Weill-Hallé, 75205 Paris, France
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2
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Fant L, Mazzarisi O, Panizon E, Grilli J. Stable cooperation emerges in stochastic multiplicative growth. Phys Rev E 2023; 108:L012401. [PMID: 37583239 DOI: 10.1103/physreve.108.l012401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2022] [Accepted: 06/01/2023] [Indexed: 08/17/2023]
Abstract
Understanding the evolutionary stability of cooperation is a central problem in biology, sociology, and economics. There exist only a few known mechanisms that guarantee the existence of cooperation and its robustness to cheating. Here, we introduce a mechanism for the emergence of cooperation in the presence of fluctuations. We consider agents whose wealth changes stochastically in a multiplicative fashion. Each agent can share part of her wealth as a public good, which is equally distributed among all the agents. We show that, when agents operate with long-time horizons, cooperation produces an advantage at the individual level, as it effectively screens agents from the deleterious effect of environmental fluctuations.
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Affiliation(s)
- Lorenzo Fant
- International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
- Instituto Gulbenkian de Ciência, Oeiras, Portugal
| | - Onofrio Mazzarisi
- Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
| | - Emanuele Panizon
- Quantitative Life Sciences section, The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34014 Trieste, Italy
| | - Jacopo Grilli
- Quantitative Life Sciences section, The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34014 Trieste, Italy
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3
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Vinod D, Cherstvy AG, Metzler R, Sokolov IM. Time-averaging and nonergodicity of reset geometric Brownian motion with drift. Phys Rev E 2022; 106:034137. [PMID: 36266856 DOI: 10.1103/physreve.106.034137] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2022] [Accepted: 08/09/2022] [Indexed: 06/16/2023]
Abstract
How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Δ)≠TAMSD(Δ) and Variance(Δ)≠TAMSD(Δ) at short lag times Δ and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Δ/T≪1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.
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Affiliation(s)
- Deepak Vinod
- Institute for Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
| | - Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
| | - Igor M Sokolov
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
- IRIS Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany
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4
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Artificial Intelligence for Sustainable Complex Socio-Technical-Economic Ecosystems. COMPUTATION 2022. [DOI: 10.3390/computation10060095] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/10/2022]
Abstract
The strong and functional couplings among ecological, economic, social, and technological processes explain the complexification of human-made systems, and phenomena such as globalization, climate change, the increased urbanization and inequality of human societies, the power of information, and the COVID-19 syndemic. Among complexification’s features are non-decomposability, asynchronous behavior, components with many degrees of freedom, increased likelihood of catastrophic events, irreversibility, nonlinear phase spaces with immense combinatorial sizes, and the impossibility of long-term, detailed prediction. Sustainability for complex systems implies enough efficiency to explore and exploit their dynamic phase spaces and enough flexibility to coevolve with their environments. This, in turn, means solving intractable nonlinear semi-structured dynamic multi-objective optimization problems, with conflicting, incommensurable, non-cooperative objectives and purposes, under dynamic uncertainty, restricted access to materials, energy, and information, and a given time horizon. Given the high-stakes; the need for effective, efficient, diverse solutions; their local and global, and present and future effects; and their unforeseen short-, medium-, and long-term impacts; achieving sustainable complex systems implies the need for Sustainability-designed Universal Intelligent Agents (SUIAs). The proposed philosophical and technological SUIAs will be heuristic devices for harnessing the strong functional coupling between human, artificial, and nonhuman biological intelligence in a non-zero-sum game to achieve sustainability.
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5
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Stojkoski V, Karbevski M. Ergodicity breaking in wealth dynamics: The case of reallocating geometric Brownian motion. Phys Rev E 2022; 105:024107. [PMID: 35291164 DOI: 10.1103/physreve.105.024107] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2021] [Accepted: 01/26/2022] [Indexed: 06/14/2023]
Abstract
A growing body of empirical evidence suggests that the dynamics of wealth within a population tends to be nonergodic, even after rescaling the individual wealth with the population average. Despite these discoveries, the way in which nonergodicity manifests itself in models of economic interactions remains an open issue. Here we shed valuable insight on these properties by studying the nonergodicity of the population average wealth in a simple model for wealth dynamics in a growing and reallocating economy called reallocating geometric Brownian motion (RGBM). When the effective wealth reallocation in the economy is from the poor to the rich, the model allows for the existence of negative wealth within the population. In this work, we show that in the negative reallocation regime of RGBM, ergodicity breaks as the difference between the time-average and the ensemble growth rate of the average wealth in the population. In particular, the ensemble average wealth grows exponentially, whereas the time-average growth rate is nonexistent. Moreover, we find that the system is characterized with a critical self-averaging time period. Before this time period, the ensemble average is a fair approximation for the population average wealth. Afterwards, the nonergodicity forces the population average to oscillate between positive and negative values since the magnitude of this observable is determined by the most extreme wealth values in the population. This implies that the dynamics of the population average is an unstable phenomenon in a nonergodic economy. We use this result to argue that one should be cautious when interpreting economic well-being measures that are based on the population average wealth in nonergodic economies.
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Affiliation(s)
- Viktor Stojkoski
- Faculty of Economics, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, 1000 Skopje, Macedonia
- Center for Collective Learning, ANITI, University of Toulouse, 31000 Toulouse, France
| | - Marko Karbevski
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, 1000 Skopje, Macedonia
- UFR de Mathématiques, Polar Cape Consulting, Sankt Eriksgatan 63b 11234 Stockholm, Sweden
- Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
- Institute of Mathematics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia
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6
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Geometric Brownian Motion (GBM) of Stock Indexes and Financial Market Uncertainty in the Context of Non-Crisis and Financial Crisis Scenarios. MATHEMATICS 2022. [DOI: 10.3390/math10030309] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
The present article proposes a methodology for modeling the evolution of stock market indexes for 2020 using geometric Brownian motion (GBM), but in which drift and diffusion are determined considering two states of economic conjunctures (states of the economy), i.e., non-crisis and financial crisis. Based on this approach, we have found that the GBM proved to be a suitable model for making forecasts of stock market index values, as it describes quite well their future evolution. However, the model proposed by us, modified geometric Brownian motion (mGBM), brings some contributions that better describe the future evolution of stock indexes. Evidence in this regard was provided by analyzing the DAX, S&P 500, and SHANGHAI Composite stock indexes. Throughout the research, it was also found that the entropy of these markets, analyzed in the periods of non-crisis and financial crisis, does not differ significantly for DAX—German Stock Exchange (EU) and S&P 500—New York Stock Exchange (US), and insignificant differences for SHANGHAI Composite—Shanghai Stock Exchange (Asia). Given the fact that there is a direct link between market efficiency and their entropy (high entropy—high efficiency; low entropy—low efficiency), it can be deduced that the analyzed markets are information-efficient in both economic conjunctures, and, in this case, the use of GBM for forecasting is justified, as the prices have a random evolution (random walk).
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7
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Albers T, Radons G. Nonergodicity of d-dimensional generalized Lévy walks and their relation to other space-time coupled models. Phys Rev E 2022; 105:014113. [PMID: 35193310 DOI: 10.1103/physreve.105.014113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2021] [Accepted: 12/23/2021] [Indexed: 06/14/2023]
Abstract
We investigate the nonergodicity of the generalized Lévy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)PRLTAO0031-900710.1103/PhysRevLett.58.1100] with respect to the squared displacements. We present detailed analytical derivations of our previous findings outlined in a recent letter [Phys. Rev. Lett. 120, 104501 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.104501], give detailed interpretations, and in particular emphasize three surprising results. First, we find that the mean-squared displacements can diverge for a certain range of parameter values. Second, we show that an ensemble of trajectories can spread subdiffusively, whereas individual time-averaged squared displacements show superdiffusion. Third, we recognize that the fluctuations of the time-averaged squared displacements can become so large that the ergodicity breaking parameter diverges, what we call infinitely strong ergodicity breaking. This phenomenon can also occur for paramter values where the lag-time dependence of the mean-squared displacements is linear indicating normal diffusion. In order to numerically determine the full distribution of time-averaged squared displacements, we use importance sampling. For an embedding of our findings into existing results in the literature, we define a more general model which we call variable speed generalized Lévy walk and which includes well-known models from the literature as special cases such as the space-time coupled Lévy flight or the anomalous Drude model. We discuss and interpret our findings regarding the generalized Lévy walk in detail and compare them with the nonergodicity of the other space-time coupled models following from the more general model.
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Affiliation(s)
- Tony Albers
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany and Institute of Mechatronics, 09126 Chemnitz, Germany
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8
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Vinod D, Cherstvy AG, Wang W, Metzler R, Sokolov IM. Nonergodicity of reset geometric Brownian motion. Phys Rev E 2022; 105:L012106. [PMID: 35193263 DOI: 10.1103/physreve.105.l012106] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2021] [Accepted: 01/10/2022] [Indexed: 06/14/2023]
Abstract
We derive. the ensemble- and time-averaged mean-squared displacements (MSD, TAMSD) for Poisson-reset geometric Brownian motion (GBM), in agreement with simulations. We find MSD and TAMSD saturation for frequent resetting, quantify the spread of TAMSDs via the ergodicity-breaking parameter and compute distributions of prices. General MSD-TAMSD nonequivalence proves reset GBM nonergodic.
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Affiliation(s)
- Deepak Vinod
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Andrey G Cherstvy
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
- Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
| | - Wei Wang
- Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
| | - Ralf Metzler
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Igor M Sokolov
- Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
- IRIS Adlershof, 12489 Berlin, Germany
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9
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Liu KKL, Lubbers N, Klein W, Tobochnik J, Boghosian BM, Gould H. Simulation of a generalized asset exchange model with economic growth and wealth distribution. Phys Rev E 2021; 104:014150. [PMID: 34412229 DOI: 10.1103/physreve.104.014150] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2021] [Accepted: 07/13/2021] [Indexed: 11/07/2022]
Abstract
The agent-based yard-sale model of wealth inequality is generalized to incorporate exponential economic growth and its distribution. The distribution of economic growth is nonuniform and is determined by the wealth of each agent and a parameter λ. Our numerical results indicate that the model has a critical point at λ=1 between a phase for λ<1 with economic mobility and exponentially growing wealth of all agents and a nonstationary phase for λ≥1 with wealth condensation and no mobility. We define the energy of the system and show that the system can be considered to be in thermodynamic equilibrium for λ<1. Our estimates of various critical exponents are consistent with a mean-field theory [see W. Klein et al., following paper, Phys. Rev. E 104, 014151 (2021)10.1103/PhysRevE.104.014151]. The exponents do not obey the usual scaling laws unless a combination of parameters that we refer to as the Ginzburg parameter is held fixed as the phase transition is approached. The model illustrates that both poorer and richer agents benefit from economic growth if its distribution does not favor the richer agents too strongly. This work and the following theoretical paper contribute to our understanding of whether the methods of equilibrium statistical mechanics can be applied to economic systems.
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Affiliation(s)
- Kang K L Liu
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - N Lubbers
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - W Klein
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA.,Center for Computational Science, Boston University, Boston, Massachusetts 02215, USA
| | - J Tobochnik
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA.,Department of Physics, Kalamazoo College, Kalamazoo, Michigan 49006, USA
| | - B M Boghosian
- Department of Mathematics, Tufts University, Medford, Massachusetts 02155, USA
| | - Harvey Gould
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA.,Department of Physics, Clark University, Worcester, Massachusetts 01610, USA
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10
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Stojkoski V, Sandev T, Kocarev L, Pal A. Geometric Brownian motion under stochastic resetting: A stationary yet nonergodic process. Phys Rev E 2021; 104:014121. [PMID: 34412255 DOI: 10.1103/physreve.104.014121] [Citation(s) in RCA: 16] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 06/16/2021] [Indexed: 01/19/2023]
Abstract
We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a canonical stochastic multiplicative process for nonstationary and nonergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains nonergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the nonergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable state, and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand-alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.
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Affiliation(s)
- Viktor Stojkoski
- Faculty of Economics, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia.,Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
| | - Trifce Sandev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia.,Institute of Physics and Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany.,Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
| | - Ljupco Kocarev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia.,Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University, P.O. Box 393, 1000 Skopje, Macedonia
| | - Arnab Pal
- School of Chemistry, The Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel
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11
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Klein W, Lubbers N, Liu KKL, Khouw T, Gould H. Mean-field theory of an asset exchange model with economic growth and wealth distribution. Phys Rev E 2021; 104:014151. [PMID: 34412228 DOI: 10.1103/physreve.104.014151] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2021] [Accepted: 07/13/2021] [Indexed: 11/07/2022]
Abstract
We develop a mean-field theory of the growth, exchange, and distribution (GED) model introduced by Liu et al. [K. K. L. Liu et al., preceding paper, Phys. Rev. E 104, 014150 (2021)10.1103/PhysRevE.104.014150] that accurately describes the phase transition in the limit that the number of agents N approaches infinity. The GED model is a generalization of the yard-sale model in which the additional wealth added by economic growth is nonuniformly distributed to the agents according to their wealth in a way determined by the parameter λ. The model is shown numerically to have a phase transition at λ=1 and be characterized by critical exponents and critical slowing down. Our mean-field treatment of the GED model correctly predicts the existence of the phase transition, a critical slowing down, and the values of the critical exponents and introduces an energy whose probability satisfies the Boltzmann distribution for λ<1, implying that the system is in thermodynamic equilibrium in the limit that N→∞. We show that the values of the critical exponents obtained by varying λ for a fixed value of N do not satisfy the usual scaling laws, but do satisfy scaling if a combination of parameters, which we refer to as the Ginzburg parameter, is much greater than one and is held constant. We discuss possible implications of our results for understanding economic systems and the subtle nature of the mean-field limit in systems with both additive and multiplicative noise.
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Affiliation(s)
- W Klein
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA and Center for Computational Science, Boston University, Boston, Massachusetts 02215, USA
| | - N Lubbers
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - Kang K L Liu
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - T Khouw
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - Harvey Gould
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA and Department of Physics, Clark University, Worcester, Massachusetts 01610, USA
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12
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Cherstvy AG, Vinod D, Aghion E, Sokolov IM, Metzler R. Scaled geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements. Phys Rev E 2021; 103:062127. [PMID: 34271619 DOI: 10.1103/physreve.103.062127] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2021] [Accepted: 05/24/2021] [Indexed: 01/21/2023]
Abstract
Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic stochastic process of geometric Brownian motion (GBM). The innate property of such models and of real stock-market prices is the roughly exponential growth of prices with time [on average, in crisis-free times]. We here explore the ensemble- and time averages of a multiplicative-noise stochastic process with power-law-like time-dependent volatility, σ(t)∼t^{α}, named scaled GBM (SGBM). For SGBM, the mean-squared displacement (MSD) computed for an ensemble of statistically equivalent trajectories can grow faster than exponentially in time, while the time-averaged MSD (TAMSD)-based on a sliding-window averaging along a single trajectory-is always linear at short lag times Δ. The proportionality factor between these the two averages of the time series is Δ/T at short lag times, where T is the trajectory length, similarly to GBM. This discrepancy of the scaling relations and pronounced nonequivalence of the MSD and TAMSD at Δ/T≪1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with σ(t)-modulation, the main focus of our analysis. The analytical predictions for the MSD and mean TAMSD for SGBM are in quantitative agreement with the results of stochastic computer simulations.
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Affiliation(s)
- Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany.,Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
| | - Deepak Vinod
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Erez Aghion
- Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
| | - Igor M Sokolov
- Institut für Physik and IRIS Adlershof, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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13
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Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing. ENTROPY 2020; 22:e22121432. [PMID: 33353060 PMCID: PMC7766185 DOI: 10.3390/e22121432] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/30/2020] [Revised: 12/11/2020] [Accepted: 12/16/2020] [Indexed: 12/13/2022]
Abstract
Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.
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14
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Stojkoski V, Utkovski Z, Basnarkov L, Kocarev L. Cooperation dynamics in networked geometric Brownian motion. Phys Rev E 2019; 99:062312. [PMID: 31330721 DOI: 10.1103/physreve.99.062312] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2018] [Indexed: 05/23/2023]
Abstract
Recent works suggest that pooling and sharing may constitute a fundamental mechanism for the evolution of cooperation in well-mixed fluctuating environments. The rationale is that, by reducing the amplitude of fluctuations, pooling and sharing increases the steady-state growth rate at which individuals self-reproduce. However, in reality interactions are seldom realized in a well-mixed structure, and the underlying topology is in general described by a complex network. Motivated by this observation, we investigate the role of the network structure on the cooperative dynamics in fluctuating environments, by developing a model for networked pooling and sharing of resources undergoing a geometric Brownian motion. The study reveals that, while in general cooperation increases the individual steady state growth rates (i.e., is evolutionary advantageous), the interplay with the network structure may yield large discrepancies in the observed individual resource endowments. We comment possible biological and social implications and discuss relations to econophysics.
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Affiliation(s)
- Viktor Stojkoski
- Academy of Sciences and Arts of the Republic of North Macedonia, P.O. Box 428, 1000 Skopje, North Macedonia
| | - Zoran Utkovski
- Fraunhofer Heinrich Hertz Institute, Einsteinufer 37, 10587, Berlin, Germany
- Faculty of Computer Science, University Goce Delcev Shtip, P.O. Box 10-A, 2000 Shtip 2000, North Macedonia
| | - Lasko Basnarkov
- Academy of Sciences and Arts of the Republic of North Macedonia, P.O. Box 428, 1000 Skopje, North Macedonia
- Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University, P.O. Box 393, 1000 Skopje, North Macedonia
| | - Ljupco Kocarev
- Academy of Sciences and Arts of the Republic of North Macedonia, P.O. Box 428, 1000 Skopje, North Macedonia
- Faculty of Computer Science, University Goce Delcev Shtip, P.O. Box 10-A, 2000 Shtip 2000, North Macedonia
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15
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Albers T, Radons G. Exact Results for the Nonergodicity of d-Dimensional Generalized Lévy Walks. PHYSICAL REVIEW LETTERS 2018; 120:104501. [PMID: 29570320 DOI: 10.1103/physrevlett.120.104501] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2017] [Indexed: 06/08/2023]
Abstract
We provide analytical results for the ensemble-averaged and time-averaged squared displacement, and the randomness of the latter, in the full two-dimensional parameter space of the d-dimensional generalized Lévy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)PRLTAO0031-900710.1103/PhysRevLett.58.1100]. In certain regions of the parameter plane, we obtain surprising results such as the divergence of the mean-squared displacements, the divergence of the ergodicity breaking parameter despite a finite mean-squared displacement, and subdiffusion which appears superdiffusive when one only considers time averages.
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Affiliation(s)
- Tony Albers
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
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16
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Chvykov P, England J. Least-rattling feedback from strong time-scale separation. Phys Rev E 2018; 97:032115. [PMID: 29776054 DOI: 10.1103/physreve.97.032115] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2017] [Indexed: 06/08/2023]
Abstract
In most interacting many-body systems associated with some "emergent phenomena," we can identify subgroups of degrees of freedom that relax on dramatically different time scales. Time-scale separation of this kind is particularly helpful in nonequilibrium systems where only the fast variables are subjected to external driving; in such a case, it may be shown through elimination of fast variables that the slow coordinates effectively experience a thermal bath of spatially varying temperature. In this paper, we investigate how such a temperature landscape arises according to how the slow variables affect the character of the driven quasisteady state reached by the fast variables. Brownian motion in the presence of spatial temperature gradients is known to lead to the accumulation of probability density in low-temperature regions. Here, we focus on the implications of attraction to low effective temperature for the long-term evolution of slow variables. After quantitatively deriving the temperature landscape for a general class of overdamped systems using a path-integral technique, we then illustrate in a simple dynamical system how the attraction to low effective temperature has a fine-tuning effect on the slow variable, selecting configurations that bring about exceptionally low force fluctuation in the fast-variable steady state. We furthermore demonstrate that a particularly strong effect of this kind can take place when the slow variable is tuned to bring about orderly, integrable motion in the fast dynamics that avoids thermalizing energy absorbed from the drive. We thus point to a potentially general feedback mechanism in multi-time-scale active systems, that leads to the exploration of slow variable space, as if in search of fine tuning for a "least-rattling" response in the fast coordinates.
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Affiliation(s)
- Pavel Chvykov
- Physics of Living Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Jeremy England
- Physics of Living Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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17
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Stollmeier F, Nagler J. Unfair and Anomalous Evolutionary Dynamics from Fluctuating Payoffs. PHYSICAL REVIEW LETTERS 2018; 120:058101. [PMID: 29481174 DOI: 10.1103/physrevlett.120.058101] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2017] [Indexed: 06/08/2023]
Abstract
Evolution occurs in populations of reproducing individuals. Reproduction depends on the payoff a strategy receives. The payoff depends on the environment that may change over time, on intrinsic uncertainties, and on other sources of randomness. These temporal variations in the payoffs can affect which traits evolve. Understanding evolutionary game dynamics that are affected by varying payoffs remains difficult. Here we study the impact of arbitrary amplitudes and covariances of temporally varying payoffs on the dynamics. The evolutionary dynamics may be "unfair," meaning that, on average, two coexisting strategies may persistently receive different payoffs. This mechanism can induce an anomalous coexistence of cooperators and defectors in the prisoner's dilemma, and an unexpected selection reversal in the hawk-dove game.
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Affiliation(s)
- Frank Stollmeier
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), Am Faßberg 17, 37077 Göttingen, Germany and Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, Am Faßberg 17, 37077 Göttingen, Germany
| | - Jan Nagler
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH 8093 Zurich, Switzerland
- Computational Social Science, Department of Humanities, Social and Political Sciences, ETH Zurich, Clausiusstrasse 50, CH-8092 Zurich, Switzerland
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18
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Patange S, Girvan M, Larson DR. Single-cell systems biology: probing the basic unit of information flow. ACTA ACUST UNITED AC 2017; 8:7-15. [PMID: 29552672 DOI: 10.1016/j.coisb.2017.11.011] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023]
Abstract
Gene expression varies across cells in a population or a tissue. This heterogeneity has come into sharp focus in recent years through developments in new imaging and sequencing technologies. However, our ability to measure variation has outpaced our ability to interpret it. Much of the variability may arise from random effects occurring in the processes of gene expression (transcription, RNA processing and decay, translation). The molecular basis of these effects is largely unknown. Likewise, a functional role of this variability in growth, differentiation and disease has only been elucidated in a few cases. In this review, we highlight recent experimental and theoretical advances for measuring and analyzing stochastic variation.
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Affiliation(s)
- Simona Patange
- Laboratory of Receptor Biology and Gene Expression, Center for Cancer Research, National Cancer Institute. Bethesda, MD 20892
- Institute for Physical Science and Technology, University of Maryland, College Park, MD
| | - Michelle Girvan
- Institute for Physical Science and Technology, University of Maryland, College Park, MD
- Department of Physics, University of Maryland. College Park, MD
| | - Daniel R Larson
- Laboratory of Receptor Biology and Gene Expression, Center for Cancer Research, National Cancer Institute. Bethesda, MD 20892
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19
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Alper O, Somekh-Baruch A, Pirvandy O, Schaps M, Yaari G. Effects of correlations and fees in random multiplicative environments: Implications for portfolio management. Phys Rev E 2017; 96:022305. [PMID: 28950518 DOI: 10.1103/physreve.96.022305] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2016] [Indexed: 11/07/2022]
Abstract
Geometric Brownian motion (GBM) is frequently used to model price dynamics of financial assets, and a weighted average of multiple GBMs is commonly used to model a financial portfolio. Diversified portfolios can lead to an increased exponential growth compared to a single asset by effectively reducing the effective noise. The sum of GBM processes is no longer a log-normal process and has a complex statistical properties. The nonergodicity of the weighted average process results in constant degradation of the exponential growth from the ensemble average toward the time average. One way to stay closer to the ensemble average is to maintain a balanced portfolio: keep the relative weights of the different assets constant over time. To keep these proportions constant, whenever assets values change, it is necessary to rebalance their relative weights, exposing this strategy to fees (transaction costs). Two strategies that were suggested in the past for cases that involve fees are rebalance the portfolio periodically and rebalance it in a partial way. In this paper, we study these two strategies in the presence of correlations and fees. We show that using periodic and partial rebalance strategies, it is possible to maintain a steady exponential growth while minimizing the losses due to fees. We also demonstrate how these redistribution strategies perform in a phenomenal way on real-world market data, despite the fact that not all assumptions of the model hold in these real-world systems. Our results have important implications for stochastic dynamics in general and to portfolio management in particular, as we show that there is a superior alternative to the common buy-and-hold strategy, even in the presence of correlations and fees.
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Affiliation(s)
- Ofer Alper
- Bar-Ilan University, Ramat Gan 5290002, Israel
| | | | - Oz Pirvandy
- Bar-Ilan University, Ramat Gan 5290002, Israel
| | | | - Gur Yaari
- Bar-Ilan University, Ramat Gan 5290002, Israel
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20
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Benisty H. Simple wealth distribution model causing inequality-induced crisis without external shocks. Phys Rev E 2017; 95:052307. [PMID: 28618493 DOI: 10.1103/physreve.95.052307] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2016] [Indexed: 11/07/2022]
Abstract
We address the issue of the dynamics of wealth accumulation and economic crisis triggered by extreme inequality, attempting to stick to most possibly intrinsic assumptions. Our general framework is that of pure or modified multiplicative processes, basically geometric Brownian motions. In contrast with the usual approach of injecting into such stochastic agent models either specific, idiosyncratic internal nonlinear interaction patterns or macroscopic disruptive features, we propose a dynamic inequality model where the attainment of a sizable fraction of the total wealth by very few agents induces a crisis regime with strong intermittency, the explicit coupling between the richest and the rest being a mere normalization mechanism, hence with minimal extrinsic assumptions. The model thus harnesses the recognized lack of ergodicity of geometric Brownian motions. It also provides a statistical intuition to the consequences of Thomas Piketty's recent "r>g" (return rate > growth rate) paradigmatic analysis of very-long-term wealth trends. We suggest that the "water-divide" of wealth flow may define effective classes, making an objective entry point to calibrate the model. Consistently, we check that a tax mechanism associated to a few percent relative bias on elementary daily transactions is able to slow or stop the build-up of large wealth. When extreme fluctuations are tamed down to a stationary regime with sizable but steadier inequalities, it should still offer opportunities to study the dynamics of crisis and the inner effective classes induced through external or internal factors.
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Affiliation(s)
- Henri Benisty
- Laboratoire Charles Fabry, Institut d'Optique Graduate School, CNRS, Univ. Paris Saclay, 2 Ave Augustin Fresnel, 91127 Palaiseau Cedex, France
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21
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Budini AA. Memory-induced diffusive-superdiffusive transition: Ensemble and time-averaged observables. Phys Rev E 2017; 95:052110. [PMID: 28618554 DOI: 10.1103/physreve.95.052110] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2017] [Indexed: 06/07/2023]
Abstract
The ensemble properties and time-averaged observables of a memory-induced diffusive-superdiffusive transition are studied. The model consists in a random walker whose transitions in a given direction depend on a weighted linear combination of the number of both right and left previous transitions. The diffusion process is nonstationary, and its probability develops the phenomenon of aging. Depending on the characteristic memory parameters, the ensemble behavior may be normal, superdiffusive, or ballistic. In contrast, the time-averaged mean squared displacement is equal to that of a normal undriven random walk, which renders the process nonergodic. In addition, and similarly to Lévy walks [Godec and Metzler, Phys. Rev. Lett. 110, 020603 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.020603], for trajectories of finite duration the time-averaged displacement apparently become random with properties that depend on the measurement time and also on the memory properties. These features are related to the nonstationary power-law decay of the transition probabilities to their stationary values. Time-averaged response to a bias is also calculated. In contrast with Lévy walks [Froemberg and Barkai, Phys. Rev. E 87, 030104(R) (2013)PLEEE81539-375510.1103/PhysRevE.87.030104], the response always vanishes asymptotically.
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Affiliation(s)
- Adrián A Budini
- Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Centro Atómico Bariloche, Avenida E. Bustillo Km 9.5, (8400) Bariloche, Argentina and Universidad Tecnológica Nacional (UTN-FRBA), Fanny Newbery 111, (8400) Bariloche, Argentina
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22
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Budini AA. Inhomogeneous diffusion and ergodicity breaking induced by global memory effects. Phys Rev E 2016; 94:052142. [PMID: 27967169 DOI: 10.1103/physreve.94.052142] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2016] [Indexed: 06/06/2023]
Abstract
We introduce a class of discrete random-walk model driven by global memory effects. At any time, the right-left transitions depend on the whole previous history of the walker, being defined by an urnlike memory mechanism. The characteristic function is calculated in an exact way, which allows us to demonstrate that the ensemble of realizations is ballistic. Asymptotically, each realization is equivalent to that of a biased Markovian diffusion process with transition rates that strongly differs from one trajectory to another. Using this "inhomogeneous diffusion" feature, the ergodic properties of the dynamics are analytically studied through the time-averaged moments. Even in the long-time regime, they remain random objects. While their average over realizations recovers the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second time-averaged moment, an ergodic limit and the limit of infinite lag times do not commutate. All these effects are induced by the memory effects. A generalized Einstein fluctuation-dissipation relation is also obtained for the time-averaged moments.
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Affiliation(s)
- Adrián A Budini
- Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Centro Atómico Bariloche, Avenida E. Bustillo Km 9.5, (8400) Bariloche, Argentina and Universidad Tecnológica Nacional (UTN-FRBA), Fanny Newbery 111, (8400) Bariloche, Argentina
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23
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Budini AA. Weak ergodicity breaking induced by global memory effects. Phys Rev E 2016; 94:022108. [PMID: 27627247 DOI: 10.1103/physreve.94.022108] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2016] [Indexed: 06/06/2023]
Abstract
We study the phenomenon of weak ergodicity breaking for a class of globally correlated random walk dynamics defined over a finite set of states. The persistence in a given state or the transition to another one depends on the whole previous temporal history of the system. A set of waiting time distributions, associated to each state, sets the random times between consecutive steps. Their mean value is finite for all states. The probability density of time-averaged observables is obtained for different memory mechanisms. This statistical object explicitly shows departures between time and ensemble averages. While the residence time in each state may have a divergent mean value, we demonstrate that this condition is in general not necessary for breaking ergodicity. Hence, we conclude that global memory effects are an alternative mechanism able to induce ergodicity breaking without involving power-law statistics. Analytical and numerical calculations support these results.
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Affiliation(s)
- Adrián A Budini
- Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Centro Atómico Bariloche, Avenida E. Bustillo Km 9.5, (8400) Bariloche, Argentina and Universidad Tecnológica Nacional (UTN-FRBA), Fanny Newbery 111, (8400) Bariloche, Argentina
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24
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Caravelli F, Sindoni L, Caccioli F, Ududec C. Optimal growth trajectories with finite carrying capacity. Phys Rev E 2016; 94:022315. [PMID: 27627325 DOI: 10.1103/physreve.94.022315] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2015] [Indexed: 06/06/2023]
Abstract
We consider the problem of finding optimal strategies that maximize the average growth rate of multiplicative stochastic processes. For a geometric Brownian motion, the problem is solved through the so-called Kelly criterion, according to which the optimal growth rate is achieved by investing a constant given fraction of resources at any step of the dynamics. We generalize these finding to the case of dynamical equations with finite carrying capacity, which can find applications in biology, mathematical ecology, and finance. We formulate the problem in terms of a stochastic process with multiplicative noise and a nonlinear drift term that is determined by the specific functional form of carrying capacity. We solve the stochastic equation for two classes of carrying capacity functions (power laws and logarithmic), and in both cases we compute the optimal trajectories of the control parameter. We further test the validity of our analytical results using numerical simulations.
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Affiliation(s)
- F Caravelli
- Invenia Labs, 27 Parkside Place, Cambridge CB1 1HQ, United Kingdom
- London Institute of Mathematical Sciences, 35a South Street, London W1K 2XF, United Kingdom
| | - L Sindoni
- Invenia Labs, 27 Parkside Place, Cambridge CB1 1HQ, United Kingdom
| | - F Caccioli
- Department of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom
- Systemic Risk Centre, London School of Economics and Political Sciences, London, United Kingdom
| | - C Ududec
- Invenia Labs, 27 Parkside Place, Cambridge CB1 1HQ, United Kingdom
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25
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Muzikar P. Unlikely Fluctuations and Non-Equilibrium Work Theorems—A Simple Example. J Phys Chem B 2016; 120:5720-3. [DOI: 10.1021/acs.jpcb.6b04510] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Paul Muzikar
- Department
of Physics, Purdue University, West Lafayette, Indiana 47907, United States
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26
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Lanoiselée Y, Grebenkov DS. Revealing nonergodic dynamics in living cells from a single particle trajectory. Phys Rev E 2016; 93:052146. [PMID: 27300868 DOI: 10.1103/physreve.93.052146] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2015] [Indexed: 06/06/2023]
Abstract
We propose the improved ergodicity and mixing estimators to identify nonergodic dynamics from a single particle trajectory. The estimators are based on the time-averaged characteristic function of the increments and can thus capture additional information on the process as compared to the conventional time-averaged mean-square displacement. The estimators are first investigated and validated for several models of anomalous diffusion, such as ergodic fractional Brownian motion and diffusion on percolating clusters, and nonergodic continuous-time random walks and scaled Brownian motion. The estimators are then applied to two sets of earlier published trajectories of mRNA molecules inside live Escherichia coli cells and of Kv2.1 potassium channels in the plasma membrane. These statistical tests did not reveal nonergodic features in the former set, while some trajectories of the latter set could be classified as nonergodic. Time averages along such trajectories are thus not representative and may be strongly misleading. Since the estimators do not rely on ensemble averages, the nonergodic features can be revealed separately for each trajectory, providing a more flexible and reliable analysis of single-particle tracking experiments in microbiology.
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Affiliation(s)
- Yann Lanoiselée
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
| | - Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
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27
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Peters O, Gell-Mann M. Evaluating gambles using dynamics. CHAOS (WOODBURY, N.Y.) 2016; 26:023103. [PMID: 26931584 DOI: 10.1063/1.4940236] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Gambles are random variables that model possible changes in wealth. Classic decision theory transforms money into utility through a utility function and defines the value of a gamble as the expectation value of utility changes. Utility functions aim to capture individual psychological characteristics, but their generality limits predictive power. Expectation value maximizers are defined as rational in economics, but expectation values are only meaningful in the presence of ensembles or in systems with ergodic properties, whereas decision-makers have no access to ensembles, and the variables representing wealth in the usual growth models do not have the relevant ergodic properties. Simultaneously addressing the shortcomings of utility and those of expectations, we propose to evaluate gambles by averaging wealth growth over time. No utility function is needed, but a dynamic must be specified to compute time averages. Linear and logarithmic "utility functions" appear as transformations that generate ergodic observables for purely additive and purely multiplicative dynamics, respectively. We highlight inconsistencies throughout the development of decision theory, whose correction clarifies that our perspective is legitimate. These invalidate a commonly cited argument for bounded utility functions.
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Affiliation(s)
- O Peters
- London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, United Kingdom
| | - M Gell-Mann
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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28
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Tang Y, Yuan R, Ao P. Summing over trajectories of stochastic dynamics with multiplicative noise. J Chem Phys 2015; 141:044125. [PMID: 25084899 DOI: 10.1063/1.4890968] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We demonstrate that previous path integral formulations for the general stochastic interpretation generate incomplete results exemplified by the geometric Brownian motion. We thus develop a novel path integral formulation for the overdamped Langevin equation with multiplicative noise. The present path integral leads to the corresponding Fokker-Planck equation, and naturally generates a normalized transition probability in examples. Our result solves the inconsistency of the previous path integral formulations for the general stochastic interpretation, and can have wide applications in chemical and physical stochastic processes.
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Affiliation(s)
- Ying Tang
- Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Ruoshi Yuan
- School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Ping Ao
- Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
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29
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Pereira T, Turaev D. Exponential energy growth in adiabatically changing Hamiltonian systems. Phys Rev E 2015; 91:010901. [PMID: 25679558 DOI: 10.1103/physreve.91.010901] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2014] [Indexed: 11/07/2022]
Abstract
We show that the mixed phase space dynamics of a typical smooth Hamiltonian system universally leads to a sustained exponential growth of energy at a slow periodic variation of parameters. We build a model for this process in terms of geometric Brownian motion with a positive drift, and relate it to the steady entropy increase after each period of the parameters variation.
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Affiliation(s)
- Tiago Pereira
- Department of Mathematics, Imperial College London SW7 2AZ, United Kingdom and London Mathematical Laboratory, London WC2N 6DF, United Kingdom
| | - Dmitry Turaev
- Department of Mathematics, Imperial College London SW7 2AZ, United Kingdom and Lobachevsky University of Nizhny Novgorod, 603950 Russia
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